The following has been my impression as a biologist:
What is the general protocol to name something after somebody (some people)?
There is none. For certain things, such as names of genes or species, there is a protocol for submitting a name to the relevant databases, which is a right reserved for authors of the publication. This name can be anything you want, although certain standards are encouraged.
People often name species after their own name. For plasmids, the convention is to acronymize the name(s) of people who created the plasmid and make it the name. For genes, this would be considered tacky (the fashion seems to be to naming them after "clever" puns instead) but I'm sure you could get it to happen with enough perseverance. But there is nothing special about it being your name, because the name of the thing is arbitrary. You are allowed to give it any sort of name, your own name is just one of the (less interesting) options.
A concept may have several origins in different fields and due to several individuals.
A concept is built on another concept which already have a name and this can happen several times.
Concepts are not formally named after people. When originally published, the authors may or may not invent a term for the concept they discovered, to facilitate its discussion. It gets named after them, when the research turns out to be so seminal that everyone cites and recites it, and the authors begin using "the Smith protocol" as shorthand for "protocol described in a recent high-profile publication by Smith and colleagues (Smith et al. Nature 2012)". If it yet persists the test of time further, it may become a de facto tradition to call this the "Smith protocol", especially once textbook authors start electing to use "Smith protocol" as the canonical name in their own texts.
If an author invents a concept, is it appropriate to name it after him/herself, or he/she should wait others call it after his/her name?
The exception I name earlier notwithstanding, absolutely not. A scientist would get laughed out of the room if he tried to present something he blatantly named after himself (some subtle reference to his name, like an anagram of his first name, might be begrudgingly accepted), unless he was perhaps a famous Nobel laureate.
If he was a famous Nobel laureate, people would still laugh, they would just wait for him to leave the room first.
A concept was invented by some author "X" in long time ago, and then it has evolved to something very modern and somehow different. Should we still call it by the name "X"?
Since I assert the naming of concepts happens not through formal procedure, but as a consequence of frequent references to the original publication, then an improved "Smith protocol" may be named the "Doe-Smith protocol" or "Doe protocol (based on the Smith protocol)" or even just "Doe protocol" if Doe manages to publish a paper which provides a useful reference for the improved version, and the improvements are substantial enough that people feel the need to cite and refer to Doe's paper at least as much as Smith's paper.
If you were trying to get something named after you, the realistic strategies in biology are:
- Discover and name a new species, plasmid, gene, etc. And hope the nomenclature committee doesn't think you're being too arrogant.
- Describe a new experimental or mathematical/computational method, and fail to give it a nice name yourself.
- Write a definitive reference which synthesizes several existing ad-hoc variants of a concept into one unified theory, and fail to give it a nice name yourself.
For 1, formal procedures exist and are detailed by the agency you submit your name proposal to. For 2 and 3, you basically write the paper, and wait for everyone and their brother to start citing your landmark publication - hopefully they will talk about your research so much that the name you used will prove too cumbersome.
- The famous Southern blot is described as only "a method of transferring fragments of DNA from agarose gels to cellulose nitrate filters" in Southern, 1975. Although extensions of this method, like the Western, were important discoveries, their popularizers got a bit less glory since it turned out that geographical puns were more fun.
- Eagle's minimal essential medium is described as "a fluid medium" in Eagle, 1955.
- Okazaki fragments were not referred to as such in Sakabe, Okazaki, 1966.